UV mapped texture is distorted on flat faces
This happens due to the triangulation of the quads:

You can see how the blue lines &quot;break&quot; (and so the lines in the texture) right where the dotted red lines marking the triangulation are.
The problem is that the UV of these faces is made of squares and rectangles, but the faces in the mesh don't have those shapes anymore.
The solution should be to unwrap the mesh again.